In this work, a passive physical human-robot interaction (pHRI) controller is proposed to enhance pHRI performance in terms of precision, cognitive load and user effort, in cases partial knowledge of the task is available. Partial knowledge refers to a subspace of SE(3) determined by the desired task, generally mixing both position and orientation variables and which is mathematically approximated by parametric expressions. The proposed scheme, which utilizes the notion of virtual constraints and the prescribed performance control methodology, is proved to be passive with respect to the interaction force, while guaranteeing constraint satisfaction in all cases. The control scheme, is experimentally validated and compared with a dissipative control scheme utilizing a KUKA LWR4+ robot in a master-slave task; experiments also include an application to a robotic assembly case.